Electromagnetic Fields

Calculus 3 · Axiom Academy

Four elegant equations that unify electricity and magnetism, revealing the deep connection between vector calculus and electromagnetic phenomena. Electric and magnetic fields are vector fields — functions that attach a vector, with both magnitude and direction, to every point in space. The electric field of a charge points away from it; the magnetic field of a current circles around it. Electric field — the force per unit charge Magnetic field — the force felt by moving charge 2. The Differential Form: Divergence and Curl The differential form uses the two operators from Calculus 3 — divergence ( ) and curl ( ) — to describe the field locally , point by point. Two of the four laws are divergence laws; the other two are curl laws. Divergence of equals the charge density. Current and changing make circulate. 3. The Integral Form: Flux and Circulation The integral form relates the fields over entire surfaces and curves. The two forms are two views of the same laws, bridged by the Divergence Theorem (for the Gauss laws) and Stokes' Theorem (for Faraday and Ampère–Maxwell). Total flux out of a closed surface equals the charge enclosed; for it is always zero. Circulation around a closed curve equals the rate of flux change, or the current enclosed. 4. Divergence: Sources and Sinks Divergence measures how much a field spreads out of a point — the net flux escaping a tiny volume. Positive means a source, negative means a sink, zero means the field only passes through.

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